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From: eklhad@ihnet.UUCP (K. A. Dahlke)
Newsgroups: net.math,net.math.stat
Subject: Does a Random Integer Contain a 9?
Message-ID: <234@ihnet.UUCP>
Date: Mon, 27-May-85 10:42:12 EDT
Article-I.D.: ihnet.234
Posted: Mon May 27 10:42:12 1985
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Xref: watmath net.math:2021 net.math.stat:107

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Last week, my meandering mind came across a problem I heard several years ago.
The question put to me then: given a random positive integer,
what is the probability that its decimal expansion contains a '9'.
After a few minutes of thought, I proudly announced "1".
Being a young, untrained high school student, I was quite confident,
Especially since those around me agreed with my answer.
Looking back, I realize the problem is not well defined.
What does it mean to select an integer at random.
When the set of choices is finite, a standard interpretation is assumed.
Each of N choices is selected with a probability of 1/N.
This becomes meaningless when the set is infinite.
This problem requires another model.

Adopting one interpretation, my answer was correct.
What is the limit, as N approaches infinity,
of the probability that a randomly selected integer from 1 to N contains a '9'.
If N = 10^M, the probability is 1 - 0.9^M, which approaches 1.

Other variants actually select an integer from the infinite set.
Suppose we select integer I, with probability 1/2^I.
In other words, a geometric progression: 1 with probability 1/2,
2 with probability 1/4, 3 with probability 1/8, etc.
This one is not as easy to solve.
What is the probability that an integer so selected contains a '9'.
Feel free to try other density functions as well.
Mail or post any thoughts.
-- 

Karl Dahlke    ihnp4!ihnet!eklhad